An initial- and boundary-value problem for a model equation for propagation of long waves

An initial- and boundary-value problem for a model equation for small-amplitude long waves is shown to be well-posed. The model has the form u_t + u_x + uu_x ? vu_xx ? α²u_xxt = 0, where x? [0, 1] and t ? 0. The solution u = u(x, t) is specified at t = 0 and on the two boundaries x = 0 and x = 1. Unique classical solutions are shown to exist, which depend continuously on variations of the specified data within appropriate function classes.     

Plane structures in thermal runaway

We consider the problem

1. u _t=u _xx+e ^u whenx ∈ ?,t > 0,
2. u(x, 0) =u ₀(x) whenx ∈ ?,

whereu ₀(x) is continuous, nonnegative and bounded. Equation (1) appears as a limit case in the analysis of combustion of a one-dimensional solid fuel. It is known that solutions of (1), (2) blow-up in a finite timeT, a phenomenon often referred to as thermal runaway. In this paper we prove the existence of blow-up profiles which are flatter than those previously observed. We also derive the asymptotic profile ofu(x, T) near its blow-up points, which are shown to be isolated.     

Generalized least squares estimation of the functional multivariate linear errors-in-variables model

Estimators of the parameters of the functional multivariate linear errors-in-variables model are obtained by the application of generalized least squares to the sample matrix of mean squares and products. The generalized least squares estimators are shown to be consistent and asymptotically multivariate normal. Relationships between generalized least squares estimation of the functional model and of the structural model are demonstrated. It is shown that estimators constructed under the assumption of normal x are appropriate for fixed x.     

Lévy functional and jump process martingales

An arbitrary jump process is considered without any assumption about the jump times and allowing the jump times to have accumulation points of arbitrary order. Certain basic martingales q(t, A) and the related Lévy system describing the jumps are introduced, and a notion of quadratic integration with respect to the predictable quadratic variation <q, q> of the basic martingales is defined. If M_t is a (locally) square integrable martingale with respect to the family of σ-fields generated by the jump process it is shown that M_t can be represented as a stochastic integral with respect to the basic martingales and with an integrand (locally) square integrable with respect to <q, q>.     

The Complexity and Effectiveness of Prediction Algorithms

The problem of predicting an arbitrary sequence x₁x₂x₃ · · · is considered with x_{t + 1} being predicted from an analysis of the word x₁x₂ · · · x_t. There are no presumptions concerning the probability structure on this process. The relation between prediction effectiveness and Kolmogorov complexity is established. Then the Hausdorff dimension of sets of sequences for which effective methods of prediction exist is estimated. The prediction method which is asymptotically superior to other arbitrary ones realized by finite automata is constructed.     

Optimum designs when the observations are second-order processes

Let the process {Y(x,t) : t?T} be observable for each x in some compact set X. Assume that Y(x, t) = θ₀f₀(x)(t) + … + θ_kf_k(x)(t) + N(t) where f_i are continuous functions from X into the reproducing kernel Hilbert space H of the mean zero random process N. The optimum designs are characterized by an Elfving's theorem with $\text{R}$ the closed convex hull of the set {(φ, f(x))_H : 6φ 6_H ≤ 1, x?X}, where (·, ·)_H is the inner product on H. It is shown that if X is convex and f_i are linear the design points may be chosen from the extreme points of X. In some problems each linear functional c′θ can be optimally estimated by a design on one point x(c). These problems are completely characterized. An example is worked and some partial results on minimax designs are obtained.     

A Galerkin Approximation for the Initial-Value Problem for Linear Second-Order Differential Equations

In this paper, a Galerkin type algorithm is given for the numerical solution of L(x)=(r(t)x'(t))'-p(t)x(t)=g(t); x(a)=x_a, x'(a)=x'_a, where r (t)>f0, and Spline hat functions form the approximating basis. Using the related quadratic form, a two-step difference equation is derived for the numerical solutions. A discrete Gronwall type lemma is then used to show that the error at the node points satisfies e_k=0(h²). If e(t) is the error function on a?t?b; it is also shown (in a variety of norms) that e(t)?Ch² and e'(t)?C₁h. Test case runs are also included. A (one step) Richardson or Rhomberg type procedure is used to show that e^R_k=0(h⁴). Thus our results are comparable to Runge-Kutta with half the function evaluations.     

On the cycle structure of certain classes of nonlinear shift registers

When m = qt, g(x_t+1, x_2t+1,…, x_(q?1)t+1) is a linear combination of only odd (or only even) elementary symmetric functions, then every cycle of the nonlinear shift register with feedback function f(x₁, x₂,…, x_m) = x₁ + g(x_t+1, x_2t+1,…, x_(q?1)t+1) has a minimal period dividing m(q+1). It is also shown that when g is derived from a cyclic code with minimum distance ?3, every cycle of this shift register has a minimal period dividing m(q + 1).     

WAVELET ESTIMATION FOR JUMPS IN A HETEROSCEDASTIC REGRESSION MODEL

11砒roductlonAnalysis ofjumps Is very important Inpractlce．Thejumps often predicts that the in－vestlgated objects are subject to sudden山auges In山aractenstlcs．刊r exaxnple,the jumps ofn 6Xchs,lxge fat6 ill illAnC6 OftCh ShOW th6 lllfiU6DC6 of th6 11POTts;llt 6y6llts h th6 WOTld Oil6nance markt;thejumps ofa seismic signal In oil exploration m叫 imply that there eistsbroken stratum In the expfored strata．It is hot 6My to d6t6Ct th6 JllthPS SlllC6 th6 llld6Ylying Signal Is Oft6l…     

Efficient simulation of finite horizon problems in queueing and insurance risk

Let ψ(u,t) be the probability that the workload in an initially empty M/G/1 queue exceeds u at time t<∞, or, equivalently, the ruin probability in the classical Crámer-Lundberg model. Assuming service times/claim sizes to be subexponential, various Monte Carlo estimators for ψ(u,t) are suggested. A key idea behind the estimators is conditional Monte Carlo. Variance estimates are derived in the regularly varying case, the efficiencies are compared numerically and also the estimators are shown to have bounded relative error in some main cases. In part, also extensions to general Lévy processes are treated.     

The wavelet detection of the jump and cusp points of a regression function

1. IntroductionMuch effort has been taken to detect the change points of a noise contaminated signal. Detection of change points is very useful in dealing with practical problems such assignal analysis, image processing and phonetic identification. For example, in dealing withelect ro encep halogr am signal ? do ct ors of t en need t o find re al sharp cusp s which exhibi t t heaccelerations and decelerations in the beating of hearts. The early work on detection ofthe change points of a regres…     

A variational approach to dislocation problems for periodic Schrödinger operators

As a simple model for lattice defects like grain boundaries in solid state physics we consider potentials which are obtained from a periodic potential V=V(x,y) on R² with period lattice Z² by setting Wt(x,y)=V(x+t,y) for x<0 and Wt(x,y)=V(x,y) for x?0, for t∈[0,1]. For Lipschitz-continuous V it is shown that the Schrödinger operators Ht=−Δ+Wt have spectrum (surface states) in the spectral gaps of H₀, for suitable t∈(0,1). We also discuss the density of these surface states as compared to the density of the bulk. Our approach is variational and it is first applied to the well-known dislocation problem (Korotyaev (2000, 2005) [15] and [16]) on the real line. We then proceed to the dislocation problem for an infinite strip and for the plane. In Appendix A, we discuss regularity properties of the eigenvalue branches in the one-dimensional dislocation problem for suitable classes of potentials.     

Nonlinear reaction-diffusion models for interacting populations

This paper proves that several initial-boundary value problems for a wide class of nonlinear reaction-diffusion equations have solutions c_i(x, t), 1 ? i ? N (with c_i(x, t) representing the concentration of the ith species at position x in a set $\text{Ω}$ at time t ? 0), which exist for all t ? 0 and are unique, smooth, nonnegative, and strictly positive for t > 0. The Volterra-Lotka predator-prey model with diffusion (to which the results above are proved to apply) is then studied in more detail. It is proved that any bounded solution of this model loses its spatial dependence and behaves like a periodic function of time alone as t → ∞. It is proved that if the spatial dimension is one or if the diffusion coefficients of the two species are equal, then all solutions are bounded.     

The Gibbs' phenomenon from a signal processing point of view

Recently, Fay and Kloppers gave two proofs to show that the well-known Gibbs' phenomenon for Fourier series at a jump discontinuity depends only on the size of the jump and is a multiple of the integral 1/π ∫₀ ^π (sin x / x) dx. We give another proof, based upon low-pass filtering of the Fourier transform, that uses the observation that a truncated Fourier series for a function ? (x) is ‘very nearly’ equal to the convolution integral 1/π ∫ _-∞ ^+∞ ? (x - t)(sin nt / t) dt.     

On the numerical characterization of the reachability cone for an essentially nonnegative matrix

Given an n×n real matrix A with nonnegative off-diagonal entries, the solution to , x₀=x(0), t?0 is x(t)=e^tAx₀. The problem of identifying the initial points x₀ for which x(t) becomes and remains entrywise nonnegative is considered. It is known that such x₀ are exactly those vectors for which the iterates x^(k)=(I+hA)^kx₀ become and remain nonnegative, where h is a positive, not necessarily small parameter that depends on the diagonal entries of A. In this paper, this characterization of initial points is extended to a numerical test when A is irreducible: if x^(k) becomes and remains positive, then so does x(t); if x(t) fails to become and remain positive, then either x^(k) becomes and remains negative or it always has a negative and a positive entry. Due to round-off errors, the latter case manifests itself numerically by x^(k) converging with a relatively small convergence ratio to a positive or a negative vector. An algorithm implementing this test is provided, along with its numerical analysis and examples. The reducible case is also discussed and a similar test is described.     

Interval criteria for oscillation of second-order half-linear differential equations

By employing a generalized Riccati technique and an integral averaging method, interval oscillation criteria are established for the second-order half-linear differential equation [r(t)|x′(t)|^α−1x′(t)]′+q(t)|x(t)|^α−1x(t)=0. These criteria are different from most known ones in the sense that they are based on information only on a sequence of subintervals of [t₀,∞), rather than on the whole half-line. They also extend, improve, and complement a number of existing results, and can be applied to extreme cases such as . In particular, several interesting examples that illustrate the importance of our results are included.     

Detection of the jump points of a heteroscedastic regression model by wavelets

1. IntroductionDetection of jump points often arises in many practical problems such as signal analysis,.... fimage processing, seismic exploratioll and phonetic identification. FOr examPle, financialeconollilsts often wad to know if abrupt changes occur in an exchange rate series sincethese changes edicted, are affecting or will affect fin-ancial market; engineers concern abolltwhether there exist jumps in a seismic signal in oil exploration bacause these jumps maypredict that there exists br…     

Navigation and control of the motion of a riderless bicycle by using a simplified dynamic model

This work deals with the guidance and control of a riderless bicycle (see Figure 1). Given two points P₁ and p₂ in the horizontal plane and a finite time interval [0, t_f]. Denote by (x₁,y₁,z₁) the coordinates of the center of the bicycle's rear wheel. Based on a simplified dynamical model of the bicycle, and by using the concept of path controllability, control laws are derived for the bicycle's pedalling moment and directional moment such that (x₁,y₁) will move from p₁ to P₂ during the time interval [0, t_f].     

Kinematical conservation laws in a space of arbitrary dimensions

In a large number of physical phenomena, we find propagating surfaces which need mathematical treatment. In this paper, we present the theory of kinematical conservation laws (KCL) in a space of arbitrary dimensions, i.e., d-D KCL, which are equations of evolution of a moving surface Ω_t in d-dimensional x-space, where x = (x ₁, x ₂,..., x _d) ∈ R^d. The KCL are derived in a specially defined ray coordinates (ξ = (ξ₁, ξ₂,..., ξ_d?1), t), where ξ₁, ξ₂,..., ξ_d?1 are surface coordinates on Ω_t and t is time. KCL are the most general equations in conservation form, governing the evolution of Ω_t with physically realistic singularities. A very special type of singularity is a kink, which is a point on Ω_t when Ω_t is a curve in R² and is a curve on Ω_t when it is a surface in R³. Across a kink the normal n to Ω_t and normal velocity m on Ω_t are discontinuous.